A212382 -id:A212382 - cp's OEIS frontend

A101785 G.f. satisfies: A(x) = 1 + xA(x)/(1 - x^2A(x)^2).

1, 1, 1, 2, 5, 12, 30, 79, 213, 584, 1628, 4600, 13138, 37871, 110043, 321978, 947813, 2805104, 8341608, 24912004, 74686460, 224694128, 678143656, 2052640752, 6229616730, 18952875247, 57792705415, 176596786934, 540679385663

Offset: 0

Paul D. Hanna, Dec 16 2004

nonn

Formula may be derived using the Lagrange Inversion theorem (cf. A049124).
a(n) = number of Dyck n-paths (A000108) all of whose descents have odd length. For example, a(3) counts UUUDDD, UDUDUD. - David Callan, Jul 25 2005
The number of noncrossing partitions of [n] with all blocks of odd size. E.g.: a(4)=5 with the five partitions being 123/4, 124/3, 134/2,1/234 and 1/2/3/4. - Louis Shapiro, Jan 07 2006
Number of ordered trees with n edges in which every non-leaf vertex has an odd number of children. - David Callan, Mar 30 2007
Number of valid hook configurations of permutations of [n] that avoid the patterns 312 and 321. - Colin Defant, Apr 28 2019

			Generated from Fibonacci polynomials (A011973) and
coefficients of odd powers of 1/(1-x):
a(1) = 1*1/1
a(2) = 1*1/1 + 0*1/3
a(3) = 1*1/1 + 1*3/3
a(4) = 1*1/1 + 2*6/3 + 0*1/5
a(5) = 1*1/1 + 3*10/3 + 1*5/5
a(6) = 1*1/1 + 4*15/3 + 3*15/5 + 0*1/7
a(7) = 1*1/1 + 5*21/3 + 6*35/5 + 1*7/7
a(8) = 1*1/1 + 6*28/3 + 10*70/5 + 4*28/7 + 0*1/9
This process is equivalent to the formula:
a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1,k)*C(n,2*k)/(2*k+1).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Vaclav Kotesovec, Asymptotic of subsequences of A212382
Helmut Prodinger, Partial Dyck path interpretation for three sequences in the Encyclopedia of Integer Sequences, arXiv:2408.01290 [math.CO], 2024.

Cf. A011973, A049124.

Column k=2 of A212382.

[n eq 0 select 1 else (&+[Binomial(n-k-1,k)*Binomial(n, 2*k)/(2*k+1): k in [0..Floor((n-1)/2)]]): n in [0..30]]; // G. C. Greubel, May 03 2019

Flatten[{1,Table[Sum[Binomial[n-k-1,k]*Binomial[n,2*k]/(2*k+1),{k,0,Floor[(n-1)/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Sep 17 2013 *)
CoefficientList[InverseSeries[Series[x*(1-x^2)/(1+x-x^2), {x, 0, 30}], x]/x, x] (* G. C. Greubel, May 03 2019 *)

{a(n)=if(n==0,1,sum(k=0,(n-1)\2,binomial(n-k-1,k)*binomial(n,2*k)/(2*k+1)))}
for(n=1, 40, print1(a(n), ", "))

N=66; Vec(serreverse(x/(1+sum(k=1,N,x^(2*k-1)))+O(x^N))/x) /* Joerg Arndt, Aug 19 2012 */

[1]+[sum(binomial(n-k-1, k)*binomial(n, 2*k)/(2*k+1) for k in (0..floor((n-1)/2))) for n in (1..30)] # G. C. Greubel, May 03 2019

a(n) = Sum_{k=0..[(n-1)/2]} C(n-k-1, k)*C(n, 2*k)/(2*k+1) for n>0, with a(0)=1.
G.f.: (1/x) * Series_Reversion( x*(1-x^2)/(1+x-x^2) ).
Recurrence: 4*n*(n+1)*(91*n^2 - 379*n + 360)*a(n) = 6*n*(182*n^3 - 849*n^2 + 1075*n - 264)*a(n-1) - 2*(182*n^4 - 1122*n^3 + 2011*n^2 - 603*n - 648)*a(n-2) + 6*(n-3)*(364*n^3 - 1698*n^2 + 2267*n - 696)*a(n-3) - 5*(n-4)*(n-3)*(91*n^2 - 197*n + 72)*a(n-4). - Vaclav Kotesovec, Sep 17 2013
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 3/4 + 1/(4*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) + 1/2*sqrt(19/6 + 76/(3*(4103 + 273*sqrt(273))^(1/3)) - 1/6*(4103 + 273*sqrt(273))^(1/3) + 63/2*sqrt(3/(19 - 304/(4103 + 273*sqrt(273))^(1/3) + 2*(4103 + 273*sqrt(273))^(1/3)))) = 3.228704951094501729... is the root of the equation 5 - 24*d + 4*d^2 - 12*d^3 + 4*d^4 = 0 and c = 0.82499074317860885542266460957609663272... is the root of the equation -125 - 3376*c^2 - 22080*c^4 - 23296*c^6 + 93184*c^8 = 0. - Vaclav Kotesovec, added Sep 17 2013, updated Jan 04 2014
G.f.: 1/(9*(3-3*x+x^2))*(x^2+27- x^2*(2*x+3)^3*(x-6)^3/(9*(3-3*x+x^2)^3*S(0) - x^2*(2*x+3)^2*(x-6)^2 )), where S(k) = 4*k+3 - x^2*(2*x^2-9*x-18)^2*(3*k+4)*(6*k+5)/( 18*(4*k+5)*(3-3*x+x^2)^3 - x^2*(2*x^2-9*x-18)^2*(3*k+5)*(6*k+7)/S(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Dec 26 2013

A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 8, 29, 85, 211, 464, 943, 1873, 3914, 9101, 23298, 61915, 162283, 409888, 996456, 2360486, 5555333, 13244114, 32357022, 80958851, 205389082, 522000262, 1317987172, 3297123652, 8190326857, 20302864970, 50482613327, 126318440989

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

The radius of convergence of g.f. A(x) is r = 5*(1-2*s+s^2)/(s*(5*s-4)) = 0.3804593157188..., where s = A(r) is described below. - Vaclav Kotesovec, Mar 20 2014

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(6) = 2: UDUDUDUDUDUD, UUUUUUDDDDDD.
a(7) = 8: UDUDUDUDUDUDUD, UDUUUUUUDDDDDD, UUUUUUDDDDDDUD, UUUUUUDDDDDUDD, UUUUUUDDDDUDDD, UUUUUUDDDUDDDD, UUUUUUDDUDDDDD, UUUUUUDUDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 10)
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=5 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^5), A), x, n+1), x, n):
seq(a(n), n=0..40);

b[x_, y_, u_] := b[x, y, u] = If[x<0 || yJean-François Alcover, Feb 05 2015, after Alois P. Heinz *)

a(n):=sum(binomial(4*k-3*n-1, n-k)*binomial(n+1, 5*k-4*n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

a(n) = sum(k=0, n, binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1); \\ Michel Marcus, Mar 05 2016

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^5).
Representation in terms of special values of generalized hypergeometric function of type 12F11: a(n) = hypergeom([1/7, 2/7, 3/7, 4/7, 5/7, 6/7, -(1/6)*n, -(1/6)*n+5/6, -(1/6)*n+2/3, -(1/6)*n+1/2, -(1/6)*n+1/3, 1/6-(1/6)*n], [1/6, 1/3, 1/3, 1/2, 1/2, 2/3, 2/3, 5/6, 5/6, 1, 7/6], 7^7/6^6), n>=0. - Karol A. Penson, Jun 21 2013
a(n) ~ s^(n+3/2) * (5*s-4)^(n+2) / (2 * sqrt(Pi) * sqrt(3*s-2) * n^(3/2) * 5^(n+5/2) * (s-1)^(2*n+9/2)), where s = 1.87696911628429... is the root of the equation 2869 - 29970*s + 138225*s^2 - 373000*s^3 + 655625*s^4 - 787500*s^5 + 656250*s^6 - 375000*s^7 + 140625*s^8 - 31250*s^9 + 3125*s^10 = 0. - Vaclav Kotesovec, Mar 20 2014
a(n) = Sum_{k=0..n}(binomial(4*k-3*n-1,n-k)*binomial(n+1,5*k-4*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016

A212383 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 3).

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 16, 37, 83, 199, 512, 1343, 3488, 9011, 23488, 62094, 165738, 444160, 1193146, 3216436, 8709766, 23683846, 64611879, 176730460, 484593740, 1332018207, 3669981318, 10133197561, 28032766982, 77688769031, 215665451243, 599644845226

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(2) = 1: UDUD.
a(3) = 1: UDUDUD.
a(4) = 2: UDUDUDUD, UUUUDDDD.
a(5) = 6: UDUDUDUDUD, UDUUUUDDDD, UUUUDDDDUD, UUUUDDDUDD, UUUUDDUDDD, UUUUDUDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=3 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^3), A), x, n+1), x, n):
seq(a(n), n=0..40);

For[A = 1; n = 1, n <= 32, n++, A = (1-(x*A)^3)/(1-x-(x*A)^3) + O[x]^n]; CoefficientList[A, x] (* Jean-François Alcover, Apr 23 2016 *)

a(n):=sum(binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

a(n) = sum(k=0, n, binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n)) /(n+1); \\ Michel Marcus, Mar 05 2016

x='x+O('x^66);  Vec( serreverse( x/(1+x/(1-x^3)) ) / x ) \\ Joerg Arndt, Apr 23 2016

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^3).
Recurrence: 27*(n-2)*n*(n+1)*(7315*n^5 - 103740*n^4 + 581321*n^3 - 1621860*n^2 + 2283420*n - 1319472)*a(n) = 18*n*(43890*n^7 - 732165*n^6 + 5094566*n^5 - 19117342*n^4 + 41486098*n^3 - 51106565*n^2 + 31574358*n - 6550776)*a(n-1) - 6*(197505*n^8 - 3591000*n^7 + 27966247*n^6 - 121632688*n^5 + 321488424*n^4 - 523324472*n^3 + 503478200*n^2 - 254789904*n + 49688640)*a(n-2) + 3*(395010*n^8 - 7774515*n^7 + 66666259*n^6 - 325204119*n^5 + 983833331*n^4 - 1877089982*n^3 + 2181330624*n^2 - 1388626960*n + 361350720)*a(n-3) + 3*(n-4)*(460845*n^7 - 7918155*n^6 + 56450863*n^5 - 217083355*n^4 + 489279096*n^3 - 653902178*n^2 + 488647076*n - 160632720)*a(n-4) - 3*(n-5)*(n-4)*(43890*n^6 - 600495*n^5 + 3490586*n^4 - 10880637*n^3 + 18337012*n^2 - 14599068*n + 3573072)*a(n-5) - 23*(n-6)*(n-5)*(n-4)*(7315*n^5 - 67165*n^4 + 239511*n^3 - 427187*n^2 + 405278*n - 173016)*a(n-6). - Vaclav Kotesovec, Aug 18 2013
a(n) ~ c*d^n/n^(3/2), where d = 2.917020449755... is the root of the equation 23 + 18*d - 189*d^2 - 162*d^3 + 162*d^4 - 108*d^5 + 27*d^6 = 0 and c = 0.415028509255451481644332... - Vaclav Kotesovec, Aug 18 2013
a(n) = Sum_{k=0..n} binomial(2*k-n-1,n-k)*binomial(n+1,3*k-2*n)/(n+1). - Vladimir Kruchinin, Mar 05 2016
G.f. is series_reversion(B(x))/x where B(x) = x/(1 + x + x^4 + x^7 + x^10 + ...) = x/(1+x/(1-x^3)); for Dyck paths with allowed ascent lengths {u_1, u_2, ...} use B(x) = x/( 1 + sum(k>=1, x^{u_k} ) ). - Joerg Arndt, Apr 23 2016

A323229 a(n) = binomial(2*n, n+1) + 1.

Original entry on oeis.org

1, 2, 5, 16, 57, 211, 793, 3004, 11441, 43759, 167961, 646647, 2496145, 9657701, 37442161, 145422676, 565722721, 2203961431, 8597496601, 33578000611, 131282408401, 513791607421, 2012616400081, 7890371113951, 30957699535777, 121548660036301, 477551179875953

Offset: 0

Peter Luschny, Feb 12 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A323230 (d=0), A260878 (d=1), this sequence (d=2).

Cf. A212382.

[Binomial(2*n, n+1) + 1: n in [0..30]]; // G. C. Greubel, Dec 26 2021

aList := proc(len) local gf, ser; assume(Im(x) > 0);
gf := (1-3*x)/(2*(x-1)*x) + (I*(1-2*x))/(2*x*sqrt(4*x-1));
ser := series(gf, x, len+2):
seq(coeff(ser, x, n), n=0..len) end: aList(27);

Table[Binomial[2n, n+1] + 1, {n, 0, 26}]

[binomial(2*n, n+1) + 1 for n in (0..30)] # G. C. Greubel, Dec 26 2021

Let G(x) = (1-3*x)/(2*(x-1)*x) + (I*(1-2*x))/(2*x*sqrt(4*x-1)) with Im(x) > 0, then a(n) = [x^n] G(x). The generating function G(x) satisfies the differential equation 6*x^3 - 4*x + 1 = (8*x^5 - 22*x^4 + 21*x^3 - 8*x^2 + x)*diff(G(x), x) + (4*x^4 - 14*x^3 + 17*x^2 - 8*x + 1)*G(x).

a(n) = A212382(2*n, n). - Alois P. Heinz, May 03 2019

A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 22, 57, 128, 268, 573, 1343, 3434, 9038, 23374, 58649, 144400, 355992, 892336, 2280020, 5892301, 15253305, 39347067, 101177783, 260255812, 671941182, 1743500452, 4542147622, 11858732144, 30983904244, 80982376879, 211831943129, 554905957520

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

The radius of convergence of g.f. A(x) is r = 4*(1-2*s+s^2)/(s*(4*s-3)) = 0.36467312501521477251..., where s = A(r) is described below. - Vaclav Kotesovec, Mar 21 2014

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(5) = 2: UDUDUDUDUD, UUUUUDDDDD.
a(6) = 7: UDUDUDUDUDUD, UDUUUUUDDDDD, UUUUUDDDDDUD, UUUUUDDDDUDD, UUUUUDDDUDDD, UUUUUDDUDDDD, UUUUUDUDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Recurrence (of order 8)
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=4 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^4), A), x, n+1), x, n):
seq(a(n), n=0..40);

a[n_] := Sum[Binomial[3k-2n-1, n-k]*Binomial[n+1, 4k-3n], {k, 0, n}]/(n+1);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)

a(n):=sum(binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n),k,0,n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

a(n) = sum(k=0, n, binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1); \\ Michel Marcus, Mar 05 2016

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^4).
a(n) ~ (s*(4*s-3))^(n+3/2) / (sqrt(Pi) * sqrt(5*s-3) * n^(3/2) * 2^(2*n+9/2) * (s-1)^(2*n+7/2)), where s = 1.880470225526517115847397... is the root of the equation 283 - 2156*s + 7312*s^2 - 14400*s^3 + 17920*s^4 - 14336*s^5 + 7168*s^6 - 2048*s^7 + 256*s^8 = 0. - Vaclav Kotesovec, Mar 21 2014
a(n) = Sum_{k=0..n} (binomial(3*k-2*n-1,n-k)*binomial(n+1,4*k-3*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016

A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 9, 37, 121, 331, 793, 1718, 3454, 6646, 12841, 26589, 61813, 158918, 426401, 1134431, 2914055, 7171539, 16967745, 39008002, 88529366, 202057561, 471422866, 1133448790, 2799775102, 7026467132, 17684574313, 44192085565, 109081884957

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(7) = 2: UDUDUDUDUDUDUD, UUUUUUUDDDDDDD.
a(8) = 9: UDUDUDUDUDUDUDUD, UDUUUUUUUDDDDDDD, UUUUUUUDDDDDDDUD, UUUUUUUDDDDDDUDD, UUUUUUUDDDDDUDDD, UUUUUUUDDDDUDDDD, UUUUUUUDDDUDDDDD, UUUUUUUDDUDDDDDD, UUUUUUUDUDDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=6 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^6), A), x, n+1), x, n):
seq(a(n), n=0..40);

a[n_] := Sum[Binomial[5k-4n-1, n-k]*Binomial[n+1, 6k-5n], {k, 0, n}]/(n+1);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)

a(n):=sum(binomial(5*k-4*n-1, n-k)*binomial(n+1, 6*k-5*n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^6).
a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 6 and r = 0.3925132712580446244..., s = 1.876653786643058101... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014
a(n) = Sum_{k=0..n} (binomial(5*k-4*n-1,n-k)*binomial(n+1,6*k-5*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016

A212387 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 7).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 10, 46, 166, 496, 1288, 3004, 6437, 12895, 24583, 45799, 87211, 180235, 420547, 1087220, 2941931, 7927664, 20705636, 51886966, 124660576, 288445186, 648173927, 1431655546, 3156274456, 7062245781, 16256654077, 38704049941, 94853117381

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(8) = 2: UDUDUDUDUDUDUDUD, UUUUUUUUDDDDDDDD.
a(9) = 10: UDUDUDUDUDUDUDUDUD, UDUUUUUUUUDDDDDDDD, UUUUUUUUDDDDDDDDUD, UUUUUUUUDDDDDDDUDD, UUUUUUUUDDDDDDUDDD, UUUUUUUUDDDDDUDDDD, UUUUUUUUDDDDUDDDDD, UUUUUUUUDDDUDDDDDD, UUUUUUUUDDUDDDDDDD, UUUUUUUUDUDDDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=7 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^7), A), x, n+1), x, n):
seq(a(n), n=0..40);

a[n_] := Sum[Binomial[6k-5n-1, n-k]*Binomial[n+1, 7k-6n], {k, 0, n}]/(n+1);
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 03 2017, after Vladimir Kruchinin *)

a(n):=sum(binomial(6*k-5*n-1, n-k)*binomial(n+1, 7*k-6*n), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 05 2016 */

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^7).
a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 7 and r = 0.4020785148135889828..., s = 1.877947072112206660... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014
a(n) = Sum_{k=0..n} (binomial(6*k-5*n-1,n-k)*binomial(n+1,7*k-6*n))/(n+1). - Vladimir Kruchinin, Mar 05 2016

A212388 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 8).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 11, 56, 221, 716, 2003, 5006, 11441, 24312, 48648, 92721, 170811, 311886, 589590, 1220979, 2864973, 7450852, 20309628, 55305706, 146505451, 373452808, 913836082, 2150455648, 4887179761, 10794337952, 23375638064, 50219351232

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(9) = 2: UDUDUDUDUDUDUDUDUD, UUUUUUUUUDDDDDDDDD.
a(10) = 11: UDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUDDDDDDDDD, UUUUUUUUUDDDDDDDDDUD, UUUUUUUUUDDDDDDDDUDD, UUUUUUUUUDDDDDDDUDDD, UUUUUUUUUDDDDDDUDDDD, UUUUUUUUUDDDDDUDDDDD, UUUUUUUUUDDDDUDDDDDD, UUUUUUUUUDDDUDDDDDDD, UUUUUUUUUDDUDDDDDDDD, UUUUUUUUUDUDDDDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..750
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=8 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^8), A), x, n+1), x, n):
seq(a(n), n=0..40);

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^8).

a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 8 and r = 0.4098875088359862102..., s = 1.880071788712472133... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014

A212389 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 9).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 12, 67, 287, 1002, 3004, 8009, 19449, 43759, 92380, 184787, 353137, 650497, 1170632, 2110021, 3977161, 8271836, 19536661, 51111062, 140210129, 385123916, 1032218316, 2670065961, 6645249777, 15922990909, 36823807747, 82485177457

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(10) = 2: UDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUDDDDDDDDDD.
a(11) = 12: UDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUDDDDDDDDDD, UUUUUUUUUUDDDDDDDDDDUD, UUUUUUUUUUDDDDDDDDDUDD, UUUUUUUUUUDDDDDDDDUDDD, UUUUUUUUUUDDDDDDDUDDDD, UUUUUUUUUUDDDDDDUDDDDD, UUUUUUUUUUDDDDDUDDDDDD, UUUUUUUUUUDDDDUDDDDDDD, UUUUUUUUUUDDDUDDDDDDDD, UUUUUUUUUUDDUDDDDDDDDD, UUUUUUUUUUDUDDDDDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..800
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=9 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or  y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^9), A), x, n+1), x, n):
seq(a(n), n=0..40);

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^9).

a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 9 and r = 0.4164039515514120671..., s = 1.882616423435763466... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014

A212390 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 10).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 13, 79, 365, 1366, 4369, 12377, 31825, 75583, 167961, 352718, 705466, 1352585, 2501205, 4495351, 7956391, 14221936, 26802361, 56058016, 133316626, 350785307, 967683665, 2677259721, 7246005881, 18977267621, 47931495649

Offset: 0

Alois P. Heinz, May 12 2012

nonn

Lengths of descents are unrestricted.

			a(0) = 1: the empty path.
a(1) = 1: UD.
a(11) = 2: UDUDUDUDUDUDUDUDUDUDUD, UUUUUUUUUUUDDDDDDDDDDD.
a(12) = 13: UDUDUDUDUDUDUDUDUDUDUDUD, UDUUUUUUUUUUUDDDDDDDDDDD, UUUUUUUUUUUDDDDDDDDDDDUD, UUUUUUUUUUUDDDDDDDDDDUDD, UUUUUUUUUUUDDDDDDDDDUDDD, UUUUUUUUUUUDDDDDDDDUDDDD, UUUUUUUUUUUDDDDDDDUDDDDD, UUUUUUUUUUUDDDDDDUDDDDDD, UUUUUUUUUUUDDDDDUDDDDDDD, UUUUUUUUUUUDDDDUDDDDDDDD, UUUUUUUUUUUDDDUDDDDDDDDD, UUUUUUUUUUUDDUDDDDDDDDDD, UUUUUUUUUUUDUDDDDDDDDDDD.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Asymptotic of subsequences of A212382

Column k=10 of A212382.

b:= proc(x, y, u) option remember;
      `if`(x<0 or y b(n$2, true):
seq(a(n), n=0..40);
# second Maple program:
a:= n-> coeff(series(RootOf(A=1+x*A/(1-(x*A)^10), A), x, n+1), x, n):
seq(a(n), n=0..40);

G.f. satisfies: A(x) = 1+x*A(x)/(1-(x*A(x))^10).

a(n) ~ s^2 / (n^(3/2) * r^(n-1/2) * sqrt(2*Pi*p*(s-1)*(1+s/(1+p*(s-1))))), where p = 10 and r = 0.421937635689419083..., s = 1.885352542104400040... are roots of the system of equations r = p*(s-1)^2 / (s*(1-p+p*s)), (r*s)^p = (s-1-r*s)/(s-1). - Vaclav Kotesovec, Jul 16 2014

cp's OEIS Frontend

A101785 G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).

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A212385 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 5).

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A212383 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 3).

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A323229 a(n) = binomial(2*n, n+1) + 1.

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A212384 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 4).

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A212386 Number of Dyck n-paths all of whose ascents have lengths equal to 1 (mod 6).

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A101785 G.f. satisfies: A(x) = 1 + xA(x)/(1 - x^2A(x)^2).